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Consider a long, thin rod of mass and length .To calculate the average linear mass density, ¯, of this one dimensional object, we can simply divide the total mass, , by the total length, : ¯ = If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, ), we can write: = Each infinitesimal unit of mass, , is equal to the product of ...
An alternative definition: A smooth vector field on a manifold is a linear map : () such that is a derivation: () = + for all , (). [ 3 ] If the manifold M {\displaystyle M} is smooth or analytic —that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields.
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...
A phase portrait is a qualitative sketch of the differential equation's behavior that shows equilibrium solutions or fixed points and the vector field on the real line. Bifurcations describe changes in the stability or existence of fixed points as a control parameter in the system changes.
In physics, a vector field (,,) is a function that returns a vector and is defined for each point (with coordinates ,,) in a region of space. The idea of sources and sinks applies to b {\displaystyle \mathbf {b} } if it follows a continuity equation of the form
A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain, Conservative vector field, a vector field that is the gradient of a scalar potential field; Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian
The advection equation is a first-order hyperbolic partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. [1] It is derived using the scalar field's conservation law, together with Gauss's theorem, and taking the infinitesimal limit.
The density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. [2] In practice, the terms density matrix and density operator are often used interchangeably.